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Last updated at Dec. 14, 2021 by Teachoo

Prove 1 + 2 + 3 + β¦β¦. + n = (π§(π§+π))/π for n, n is a natural number Step 1: Let P(n) : (the given statement) Let P(n): 1 + 2 + 3 + β¦β¦. + n = (n(n + 1))/2 Step 2: Prove for n = 1 For n = 1, L.H.S = 1 R.H.S = (π(π + 1))/2 = (1(1 + 1))/2 = (1 Γ 2)/2 = 1 Since, L.H.S. = R.H.S β΄ P(n) is true for n = 1 Step 3: Assume P(k) to be true and then prove P(k + 1) is true Assume that P(k) is true, P(k): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2 We will prove that P(k + 1) is true. P(k + 1): 1 + 2 + 3 +β¦β¦. + (k + 1) = ((k + 1)( (k + 1) + 1))/2 P(k + 1): 1 + 2 + 3 +β¦β¦.+ k + (k + 1) = ((π€ + π)(π€ + π))/π We have to prove P(k + 1) is true Solving LHS 1 + 2 + 3 +β¦β¦.+ k + (k + 1) From (1): 1 + 2 + 3 + β¦β¦. + k = (π(π + 1))/2 = (π(π + π))/π + (k + 1) = (π(π + 1) + 2(π + 1))/2 = ((π + π)(π + π))/π = RHS β΄ P(k + 1) is true when P(k) is true Step 4: Write the following line Thus, By the principle of mathematical induction, P(n) is true for n, where n is a natural number